Let $X$ be a connected compact Kähler manifold of dimension $n$ and $\theta $ be a smooth closed real $(1,1)$-form on $X$ representing a big cohomology class. Fix a model potential $\phi \in \mathrm{PSH}(X,\theta )_{>0}$.
We first study a natural metric on $\mathcal{E}^1(X,\theta ;\phi )$.
Definition 17.10.41.08EFAEA550B1 Let $\varphi ,\psi \in \mathcal{E}^1(X,\theta ;\phi )$, we define the Darvas metric $d_1$ as follows:
\[ d_1(\varphi ,\psi )=E_{\theta }^{\phi }(\varphi )+E_{\theta }^{\phi }(\psi )-2E_{\theta }^{\phi }(\varphi \land \psi ). \]
Note that by proposition 16.5.49, $\varphi \land \psi \in \mathcal{E}^1(X,\theta ;\phi )$.
Recall that $E_{\theta }^{\phi }$ is defined in definition
16.5.10.
In particular, if $\varphi \leq \psi $, we have
D8C8C0BB6585
17.10.42
\begin{equation} d_1(\varphi ,\psi )=E_{\theta }^{\phi }(\psi )-E_{\theta }^{\phi }(\varphi ). \end{equation}
We wish to show that $d_1$ is a complete metric. We first prove a contraction property:
Proposition 17.10.43.61DD221912DE Let $\varphi ,\psi ,\gamma \in \mathcal{E}^1(X,\theta ;\phi )$. Then
DCF5BF05B577
17.10.44
\begin{equation} d_1(\varphi ,\psi )\geq d_1(\varphi \land \gamma ,\psi \land \gamma ). \end{equation}
Proof.
Step 1. We first assume that $\varphi \geq \psi $. Then
\[ \begin{aligned} d_1(\varphi ,\psi )=& E_{\theta }^{\phi }(\varphi )-E_{\theta }^{\phi }(\psi )\\ \geq & E_{\theta }^{\phi }\Bigl ((\varphi \land \gamma )\lor \psi \Bigr )-E_{\theta }^{\phi }(\psi ) & \textup{by \href{../../tag/93210731F7FD/}{16.5.14}}\\ \geq & E_{\theta }^{\phi }(\varphi \land \gamma )-E_{\theta }^{\phi }(\psi \land \gamma )& \textup{by corollary \href{../../tag/4A849C02F386/}{16.5.59}}. \end{aligned} \]
Step 2. We prove the general case.
By Step 1, we have
\[ d_1(\varphi ,\varphi \land \psi )\geq d_1(\varphi \land \gamma ,\varphi \land \psi \land \gamma ),\quad d_1(\psi ,\varphi \land \psi )\geq d_1(\psi \land \gamma ,\varphi \land \psi \land \gamma ). \]
Adding the two inequalities together, we conclude (17.10.44).
$\square$
Lemma 17.10.45.488B7E9EA42C The function $d_1$ is a metric on $\mathcal{E}^1(X,\theta ;\phi )$.
Proof.
There are two facts to prove, as in the two steps below.
Step 1. Let $\varphi ,\psi \in \mathcal{E}^1(X,\theta ;\phi )$. Assume that $d_1(\varphi ,\psi )=0$. Then we will show that $\varphi =\psi $.
We first treat the case $\psi \leq \varphi $. Then $d_1(\varphi ,\psi )=E_\theta ^\phi (\varphi )-E_\theta ^\phi (\psi )=0$, and proposition 16.5.30 gives
\[ \int _ X (\varphi -\psi )\, \theta _{\psi }^ n=0. \]
We conclude that $\psi =\varphi $ using the domination principle theorem 13.13.43.
In general, put $\gamma =\varphi \land \psi $. Since
\[ d_1(\varphi ,\psi )=d_1(\varphi ,\gamma )+d_1(\psi ,\gamma ), \]
both terms on the right vanish. Applying the comparable case to the pairs $(\varphi ,\gamma )$ and $(\psi ,\gamma )$ gives $\varphi =\gamma =\psi $.
Step 2. Let $\varphi ,\psi ,\gamma \in \mathcal{E}^1(X,\theta ;\phi )$. We prove the triangle inequality:
\[ d_1(\varphi ,\psi )\leq d_1(\varphi ,\gamma )+d_1(\psi ,\gamma ). \]
This can be translated to
\[ E_{\theta }^{\phi }(\varphi \land \gamma )-E_{\theta }^{\phi }(\varphi \land \psi )\leq E_{\theta }^{\phi }(\gamma )-E_{\theta }^{\phi }(\psi \land \gamma ). \]
We just have to compute using proposition 17.10.43:
\[ \begin{aligned} E_{\theta }^{\phi }(\gamma )-E_{\theta }^{\phi }(\psi \land \gamma )\geq & E_{\theta }^{\phi }(\varphi \land \gamma )-E_{\theta }^{\phi }(\varphi \land \psi \land \gamma )\\ \geq & E_{\theta }^{\phi }(\varphi \land \gamma )-E_{\theta }^{\phi }(\varphi \land \psi ), \end{aligned} \]
where the second line follows from 16.5.14.
$\square$
We introduce an auxiliary functional.
Definition 17.10.46.D926E3A0E83A Define $I_{\theta }\colon \mathcal{E}^1(X,\theta ;\phi )\times \mathcal{E}^1(X,\theta ;\phi )\rightarrow \mathbb {R}\cup \{ \infty \} $ as follows:
4.28
\begin{equation} I_{\theta }(\varphi ,\psi )\coloneqq \int _ X |\varphi -\psi |\left( \theta _{\varphi }^ n+\theta _{\psi }^ n \right). \end{equation}
We will soon see that $I_{\theta }$ is always finite, see theorem
17.10.48.
We observe the following elementary equality.
Lemma 17.10.47.CB87AA08E67C Let $\varphi ,\psi \in \mathcal{E}^1(X,\theta ;\phi )$. Then
\[ I_{\theta }(\varphi ,\psi )=I_{\theta }(\varphi \lor \psi ,\varphi )+I_{\theta }(\varphi \lor \psi ,\psi ). \]
Proof.
By the plurifine locality of non-pluripolar products proposition 13.11.4 1, we can write
\[ \begin{aligned} I_{\theta }(\varphi ,\psi )=& \int _{ \{ \varphi <\psi \} } (\psi -\varphi ) \left( \theta _{\varphi }^ n+\theta _{\psi }^ n \right)+\int _{ \{ \varphi >\psi \} } (\varphi -\psi ) \left( \theta _{\varphi }^ n+\theta _{\psi }^ n \right),\\ I_{\theta }(\varphi \lor \psi ,\varphi )=& \int _{\{ \psi >\varphi \} } (\psi -\varphi )\left(\theta _{\psi }^ n+\theta _{\varphi }^ n \right),\\ I_{\theta }(\varphi \lor \psi ,\psi )=& \int _{\{ \psi <\varphi \} } (\varphi -\psi )\left(\theta _{\psi }^ n+\theta _{\varphi }^ n \right). \end{aligned} \]
The desired equality follows by comparing these terms.
$\square$
We have an interesting relation between $d_1$ and $I_{\theta }$ defined in definition
17.10.46.
Theorem 17.10.48.1178F95447B1 Let $\varphi ,\psi \in \mathcal{E}^1(X,\theta ;\phi )$. Then
59E3B0356FFD
17.10.49
\begin{equation} \frac{1}{C_ n} I_{\theta }(\varphi ,\psi )\leq d_1(\varphi ,\psi )\leq I_{\theta }(\varphi ,\psi ), \end{equation}
where $C_ n=3\cdot 2^{n+2}(n+1)$.
Proof.
Step 1. We first prove the right-hand part of (17.10.49).
Thanks to proposition 16.5.36 and 16.3.3, we have
\[ \begin{aligned} E_{\theta }^{\phi }(\varphi )-E_{\theta }^{\phi }(\varphi \land \psi )\leq & \int _ X (\varphi -\varphi \land \psi )\, \theta _{\varphi \land \psi }^ n \\ \leq & \int _{\{ \psi =\varphi \land \psi \} } (\varphi -\psi )\, \theta _{\psi }^ n\\ \leq & \int _ X |\varphi -\psi |\, \theta _{\psi }^ n. \end{aligned} \]
Similarly,
\[ E_{\theta }^{\phi }(\psi )-E_{\theta }^{\phi }(\varphi \land \psi )\leq \int _ X |\varphi -\psi |\, \theta _{\varphi }^ n. \]
Adding these inequalities up, we find
\[ d_1(\varphi ,\psi )\leq I_{\theta }(\varphi ,\psi ). \]
Step 2. We prove the left-hand part of (17.10.49).
We claim that
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17.10.50
\begin{equation} d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\leq \frac{3(n+1)}{2} d_1(\varphi ,\psi ). \end{equation}
For this purpose, we compute directly
\[ \begin{aligned} & d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\\ =& d_1\left(\varphi ,\varphi \land \frac{\varphi +\psi }{2}\right)+d_1\left(\frac{\varphi +\psi }{2},\varphi \land \frac{\varphi +\psi }{2}\right)\\ \leq & d_1\left(\varphi ,\varphi \land \psi \right)+d_1\left(\frac{\varphi +\psi }{2},\varphi \land \psi \right)& \textup{by \href{../../tag/93210731F7FD/}{16.5.14}}\\ \leq & \int _ X \left( \varphi -\varphi \land \psi \right)\, \theta _{\varphi \land \psi }^ n+ \int _ X \left( \frac{\varphi +\psi }{2}-\varphi \land \psi \right)\, \theta _{\varphi \land \psi }^ n & \textup{by proposition \href{../../tag/51CFFEE97C34/}{16.5.36}}\\ =& \frac{3}{2}\int _ X \left( \varphi -\varphi \land \psi \right)\, \theta _{\varphi \land \psi }^ n+\frac{1}{2}\int _ X \left( \psi -\varphi \land \psi \right)\, \theta _{\varphi \land \psi }^ n \\ \leq & \frac{3(n+1)}{2}d_1\left(\varphi ,\varphi \land \psi \right) + \frac{n+1}{2}d_1(\psi ,\varphi \land \psi )& \textup{by (\href{../../tag/F8225060CBAC/}{16.5.31})}\\ \leq & \frac{3(n+1)}{2}d_1(\varphi ,\psi ), \end{aligned} \]
and (17.10.50) follows.
We now estimate the left-hand side of (17.10.50):
\[ \begin{aligned} d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\geq & d_1\left(\varphi ,\varphi \land \frac{\varphi +\psi }{2}\right)\\ \geq & \int _ X \left(\varphi -\varphi \land \frac{\varphi +\psi }{2}\right)\, \theta _{\varphi }^ n\\ \geq & \int _{\{ \varphi >\psi \} }\left(\varphi -\varphi \land \frac{\varphi +\psi }{2}\right)\, \theta _{\varphi }^ n \\ \geq & \int _{\{ \varphi >\psi \} }\left(\varphi -\frac{\varphi +\psi }{2}\right)\, \theta _{\varphi }^ n\\ =& \frac{1}{2}\int _{\{ \varphi >\psi \} }\left(\varphi -\psi \right)\, \theta _{\varphi }^ n, \end{aligned} \]
where the second inequality follows from proposition 16.5.36.
Similarly,
\[ \begin{aligned} d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\geq & d_1\left(\frac{\varphi +\psi }{2},\varphi \land \frac{\varphi +\psi }{2}\right)\\ \geq & \int _ X \left(\frac{\varphi +\psi }{2}-\varphi \land \frac{\varphi +\psi }{2}\right)\, \theta _{(\varphi +\psi )/2}^ n\\ \geq & 2^{-n} \int _ X \left(\frac{\varphi +\psi }{2}-\varphi \land \frac{\varphi +\psi }{2}\right)\, \theta _{\varphi }^ n\\ \geq & 2^{-n} \int _{\{ \varphi <\psi \} } \left(\frac{\varphi +\psi }{2}-\varphi \land \frac{\varphi +\psi }{2}\right)\, \theta _{\varphi }^ n\\ \geq & 2^{-n} \int _{\{ \varphi <\psi \} } \left(\frac{\varphi +\psi }{2}-\varphi \right)\, \theta _{\varphi }^ n\\ =& 2^{-n-1} \int _{\{ \varphi <\psi \} } \left(\psi -\varphi \right)\, \theta _{\varphi }^ n, \end{aligned} \]
where the third inequality follows from the multi-linearity of the non-pluripolar product proposition 13.11.4 6.
Adding these estimates up, we find
\[ 2^{n+2} d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\geq \left(2^{n+1}+2\right) d_1\left(\varphi ,\frac{\varphi +\psi }{2}\right)\geq \int _ X |\varphi -\psi |\, \theta _{\varphi }^ n. \]
Therefore, combined with (17.10.50) we conclude
\[ 3\cdot 2^{n+1}(n+1)d_1(\varphi ,\psi )\geq \int _ X |\varphi -\psi |\, \theta _{\varphi }^ n. \]
By symmetry, we also get a similar expression after exchanging $\varphi $ and $\psi $. Adding these inequalities together, we find
\[ 3\cdot 2^{n+2}(n+1)d_1(\varphi ,\psi )\geq I_{\theta }(\varphi ,\psi ). \]
The left-hand part of (17.10.49) then follows.
$\square$
Corollary 17.10.51.93BC4D45D9EF Let $\varphi ,\psi \in \mathcal{E}^1(X,\theta ;\phi )$. Then
9D4858FD76E4
17.10.52
\begin{equation} d_1(\varphi \lor \psi ,\varphi )\leq C_ n d_1(\varphi ,\psi ), \end{equation}
where $C_ n$ is the constant in theorem 17.10.48.
Proof.
By theorem 17.10.48 and 17.10.47, we have
\[ d_1(\varphi \lor \psi ,\varphi )\leq I_{\theta }(\varphi \lor \psi ,\varphi )\leq I_{\theta }(\varphi ,\psi )\leq C_ n d_1(\varphi ,\psi ). \]
This completes the proof.
$\square$
Lemma 17.10.53.B3BB9434F2D9 There is $A,B>0$ so that for any $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$, we have
FC79763AA6F5
17.10.54
\begin{equation} d_1(\varphi ,\phi )\geq -\int _ X \theta _{\phi }^ n\cdot \sup _ X \varphi \geq -Ad_1(\varphi ,\phi )-B. \end{equation}
Proof.
When $\sup _ X \varphi \leq 0$, the right-hand part of (17.10.54) is trivial. Since $\varphi \leq \phi $, we find
\[ d_1(\phi ,\varphi )=-E_{\theta }^{\phi }(\varphi )\geq -\int _ X \theta _{\phi }^ n\cdot \sup _{\{ \phi \neq -\infty \} } (\varphi -\phi )=-\int _ X \theta _{\phi }^ n\cdot \sup _ X \varphi , \]
where the last equality follows from proposition 16.3.21.
We can therefore assume that $\sup _ X \varphi > 0$. In this case, the left-hand part of (17.10.54) is trivial. Take a Kähler form $\omega \geq \theta $ on $X$. Then thanks to theorem 16.3.12, we can find a constant $C>0$ so that
\[ \theta _{\phi }^ n\leq C\omega ^ n. \]
Thanks to proposition 5.17.3, there is a constant $C'>0$, independent of the choice of $\varphi $, so that
\[ \int _ X \left( \phi -\varphi +\sup _ X \varphi \right)\, \theta _{\phi }^ n\leq C'. \]
We estimate
\[ \begin{aligned} I_{\theta }(\phi ,\varphi )\geq & \int _ X |\varphi -\phi |\, \theta _{\phi }^ n\\ \geq & \left(\sup _ X \varphi \right)\int _ X \theta _{\phi }^ n-\int _ X \left( \phi -\varphi +\sup _ X \varphi \right)\, \theta _{\phi }^ n \\ \geq & \left(\sup _ X \varphi \right)\int _ X \theta _{\phi }^ n-C'. \end{aligned} \]
The right-hand part of (17.10.54) then follows from theorem 17.10.48.
$\square$
Next we handle the completeness of $d_1$. The completeness can be proved in a more general framework.
Definition 17.10.55.3FB52FA474A3 Let $E$ be a set. A pre-rooftop structure on $E$ is a binary operator $\land :E\times E\rightarrow E$, satisfying the following axioms: For $x,y,z\in E$,
2AE911379716
$x\land y=y\land x$.
67C00B3E32DE
$(x\land y)\land z=x\land (y\land z)$.
0739848FF90B
$x\land x=x$.
We call $(E,\land )$ a pre-rooftop space.
A pre-rooftop structure $\land $ defines a partial order $\leq $ on $E$ as follows:
\[ x\leq y\quad \text{if and only if } \quad x\land y=x. \]
Here by abuse of notation, we use $\leq $ to denote the partial order.
In particular, it makes sense to talk about increasing and decreasing sequences in $E$.
Definition 17.10.56.602E107F6064 Let $(E,d)$ be a metric space. A pre-rooftop structure on $(E,d)$ is a pre-rooftop structure $\land $ on $E$. We say $(E,d,\land )$ is a pre-rooftop metric space.
A rooftop structure on $(E,d)$ is a pre-rooftop structure $\land $ on $E$ such that
8F3EC66A0258
17.10.57
\begin{equation} d(x\land z,y\land z)\leq d(x,y),\quad \forall x,y,z\in E. \end{equation}
We call $(E,d,\land )$ a rooftop metric space.
Lemma 17.10.58.DB740351ED65 Let $(E,d,\land )$ be a rooftop metric space. Let $x,y,x',y'\in E$. Then
4.61
\begin{equation} d(x\land y,x'\land y')\leq d(x,x')+d(y,y'). \end{equation}
In particular, $\land \colon E\times E\rightarrow E$ is continuous.
Proof.
We compute
\[ d(x\land y,x'\land y')\leq d(x\land y,x\land y')+d(x\land y',x'\land y')\leq d(x,x')+d(y,y'). \]
This completes the proof.
$\square$
Proposition 17.10.59.8F175566EB00 Let $(E,d,\land )$ be a rooftop metric space. Then $(E,d)$ is complete if and only if both of the following hold:
0E5A5FA43046
Each increasing Cauchy sequence converges.
535E2E2720BC
Each decreasing Cauchy sequence converges.
Proof.
The direct implication is trivial.
Conversely, assume that both conditions are true. Let $(x_ j)_{j>0}$ be a Cauchy sequence in $E$. We want to prove that $(x_ j)_ j$ converges. By passing to a subsequence, we may assume that
\[ d\left(x_ j,x_{j+1}\right)\leq 2^{-j}. \]
For $k,j\geq 1$, let
\[ y_ j^ k\coloneqq x_ k\land \cdots \land x_{k+j}. \]
Then $(y_ j^ k)_ j$ is decreasing, and
\[ d\left(y_ j^ k,y_{j+1}^ k\right)\leq d\left(x_{k+j},x_{k+j+1}\right)\leq 2^{-k-j}. \]
So $(y_ j^ k)_ j$ is a decreasing Cauchy sequence. Define
\[ y^ k\coloneqq \lim _{j\to \infty }y_ j^ k. \]
Then
\[ d\left(y^ k,y^{k+1}\right)=\lim _{j\to \infty }d\left(y_{j+1}^ k,y_ j^{k+1}\right)\leq d\left(x_ k,x_{k+1}\right)\leq 2^{-k}. \]
We also need to check that the sequence $(y^ k)_ k$ is increasing. For each $j\geq 1$, we have
\[ y_ j^ k=x_ k\land y_{j-1}^{k+1}\leq y_{j-1}^{k+1}. \]
Equivalently,
\[ y_ j^ k\land y_{j-1}^{k+1}=y_ j^ k. \]
Letting $j\to \infty $ and using the continuity of $\land $ from 17.10.58, we obtain
\[ y^ k\land y^{k+1}=y^ k. \]
Thus $y^ k\leq y^{k+1}$. So $(y^ k)_ k$ is an increasing Cauchy sequence. Let
\[ y\coloneqq \lim _{k\to \infty }y^ k. \]
For every $j\geq 0$, we have
\[ d\left(y_ j^ k,x_ k\right)\leq \sum _{r=k}^{k+j-1}d\left(x_ r,x_{r+1}\right), \]
with the empty sum understood as $0$. Indeed, this is trivial for $j=0$, and the induction step follows from
\[ y_ j^ k=x_ k\land y_{j-1}^{k+1} \]
and
\[ d\left(y_ j^ k,x_ k\right)\leq d\left(y_{j-1}^{k+1},x_ k\right)\leq d\left(y_{j-1}^{k+1},x_{k+1}\right)+d\left(x_{k+1},x_ k\right). \]
Letting $j\to \infty $, we get
\[ d\left(y^ k,x_ k\right)\leq \sum _{r=k}^{\infty }d\left(x_ r,x_{r+1}\right)\leq 2^{1-k}. \]
So $(x_ k)_ k$ converges to $y$.
$\square$
Theorem 17.10.60.C89C58354E49 The metric space $(\mathcal{E}^1(X,\theta ;\phi ),d_1)$ is complete.
It follows from the proof below that if $(\varphi _ j)_{j>0}$ is a monotone sequence in $\mathcal{E}^1(X,\theta ;\phi )$ with $d_1$-limit $\varphi $, then $\varphi $ is the almost everywhere limit of $(\varphi _ j)_{j>0}$. We will use this result without further explanation in the sequel.
Proof.
As we have seen in 17.10.45 and proposition 17.10.43, the triple
\[ \left(\mathcal{E}^1(X,\theta ;\phi ),d_1,\land \right) \]
is a rooftop metric space. Hence thanks to proposition 17.10.59, it remains to show that each increasing or decreasing Cauchy sequence in $\mathcal{E}^1(X,\theta ;\phi )$ converges.
We first consider an increasing sequence $(\varphi _ i)_{i>0}$ in $\mathcal{E}^1(X,\theta ;\phi )$. The Cauchy property simply means that
\[ \sup _{i>0} E_{\theta }^{\phi }(\varphi _ i)<\infty . \]
We claim that
\[ \varphi \coloneqq \operatorname *{sup}_{i>0}{}\! {}^{*}\varphi _ i \]
is the $d_1$-limit of the sequence. We first observe that $(\sup _ X \varphi _ i)_{i>0}$ is bounded, as a consequence of 17.10.53. Therefore, proposition 16.5.53 guarantees that $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$, and hence our assertion follows from proposition 16.5.37.
Next we consider a decreasing sequence $(\varphi _ i)_{i>0}$ in $\mathcal{E}^1(X,\theta ;\phi )$. The Cauchy property simply means that
\[ \inf _{i>0} E_{\theta }^{\phi }(\varphi _ i)>-\infty . \]
We claim that
\[ \varphi \coloneqq \inf _{i>0}\varphi _ i \]
is the $d_1$-limit of the sequence. Equivalently, we need to show that $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$ and
25452B178A81
17.10.61
\begin{equation} E_{\theta }^{\phi }(\varphi )=\lim _{i\to \infty } E_{\theta }^{\phi }(\varphi _ i). \end{equation}
We first observe that $(\sup _ X \varphi _ i)_{i>0}$ is bounded, as a consequence of 17.10.53. Therefore, $\varphi \in \mathrm{PSH}(X,\theta ;\phi )$ and $\varphi _ i\xlongrightarrow {L^1} \varphi $, see corollary 5.17.5. We claim that $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$. For this purpose, we may assume that $\varphi _1\leq 0$. Since $(\varphi _ i)_ i$ is decreasing, $\varphi _ i\leq \varphi _1\leq 0$ for all $i>0$. As $\varphi _ i\preceq \phi $ and $\phi $ is model, each $\varphi _ i$ is a candidate for $P_\theta [\phi ]=\phi $, so $\varphi _ i\leq \phi $. Thus, using the energy formula (16.5.11) and the fact that all mixed terms are non-positive, we get for each $i>0$,
\[ \frac{1}{n+1}\int _ X \left(\varphi _ i-\phi \right)\, \theta _{\varphi _ i}^ n\geq E_{\theta }^{\phi }(\varphi _ i)\geq \inf _{j>0} E_{\theta }^{\phi }(\varphi _ j)>-\infty . \]
We can then apply 16.5.45, with $\gamma =\phi $, to conclude that $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$. Our assertion (17.10.61) then follows from proposition 16.5.20. We conclude the proof.
$\square$
Lemma 17.10.62.ACEE2B318CA0 Let $(\varphi _ i)_{i>0}$ be a sequence in $\mathcal{E}^1(X,\theta ;\phi )$ with $d_1$-limit $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$. Then after replacing $(\varphi _ i)_{i>0}$ by a subsequence, we can find two sequences $(\psi _ i)_{i>0}$ and $(\eta _ i)_{i>0}$ in $\mathcal{E}^1(X,\theta ;\phi )$ such that
0150B2D3A5B1
$(\psi _ i)_{i>0}$ is decreasing with $d_1$ and pointwise limit $\varphi $;
C831223AA8E9
$(\eta _ i)_{i>0}$ is increasing with $d_1$ and almost everywhere limit $\varphi $.
DB8A55157016
$\eta _ i\leq \varphi _ i\leq \psi _ i$ for all $i>0$.
Proof.
We first note that as a consequence of 17.10.53, $(\sup _ X \varphi _ i)_{i>0}$ is bounded. Let $C_ n$ be the constant in theorem 17.10.48. After replacing $(\varphi _ i)_{i>0}$ by a subsequence, we may assume that
\[ d_1(\varphi _ i,\varphi _{i+1})\leq \min \{ C_ n^{-2i},2^{-i}\} . \]
We first construct $(\psi _ i)_{i>0}$. For this purpose, it suffices to define
\[ \psi _ i\coloneqq \operatorname *{sup}_{j\geq i}{}\! {}^{*} \varphi _ j \]
for each $i>0$. It follows from proposition 16.5.53 that $\psi _ i\in \mathcal{E}^1(X,\theta ;\phi )$ for each $i>0$. For $k\geq 0$, set
\[ \psi _ i^ k\coloneqq \varphi _ i\lor \varphi _{i+1}\lor \cdots \lor \varphi _{i+k}. \]
Iterating corollary 17.10.51 gives
\[ d_1(\varphi _ i,\psi _ i^ k)\leq \sum _{a=i}^{i+k-1}C_ n^{a+1-i}d_1(\varphi _ a,\varphi _{a+1})\leq \sum _{a=i}^{i+k-1}C_ n^{a+1-i}C_ n^{-2a}. \]
Since $\psi _ i^ k$ increases almost everywhere to $\psi _ i$, letting $k\to \infty $ and using proposition 16.5.37, we get
\[ d_1(\varphi _ i,\psi _ i)\leq \frac{C_ n^{1-2i}}{1-C_ n^{-1}}. \]
Thus $\psi _ i\to \varphi $ in $d_1$. Since $(\psi _ i)_ i$ is decreasing, the monotone case in the proof of theorem 17.10.60 shows that $\psi _ i$ converges almost everywhere to $\varphi $, hence pointwise by proposition 5.14.6.
Next we construct $(\eta _ i)_{i>0}$. For each $i>0$ and $k\geq 0$, we let
\[ \eta _ i^ k\coloneqq \varphi _ i\land \varphi _{i+1}\land \cdots \land \varphi _{i+k}. \]
Then
\[ \begin{aligned} d_1(\varphi _ i,\eta _ i^ k)\leq & \sum _{j=0}^{k-1} d_1\left(\eta _ i^ j,\eta _ i^{j+1}\right)\\ \leq & \sum _{j=0}^{k-1} d_1\left(\varphi _{i+j},\varphi _{i+j+1}\right) & \textup{by proposition \href{../../tag/61DD221912DE/}{17.10.43}}\\ \leq & \sum _{j=0}^{k-1} 2^{-i-j}\\ \leq & 2^{1-i}. \end{aligned} \]
Since $\eta _ i^ k\leq \varphi _ i$, the preceding estimate gives
\[ E_{\theta }^{\phi }(\eta _ i^ k)\geq E_{\theta }^{\phi }(\varphi _ i)-2^{1-i},\quad k\geq 0. \]
The sequence $(\eta _ i^ k)_{k\geq 0}$ is decreasing, and the monotonicity of $E_{\theta }^{\phi }$ shows that the sequence $\left(E_{\theta }^{\phi }(\eta _ i^ k)\right)_{k\geq 0}$ is decreasing and bounded from below, hence convergent. Thus, for $k\leq l$,
\[ d_1(\eta _ i^ k,\eta _ i^ l)=E_{\theta }^{\phi }(\eta _ i^ k)-E_{\theta }^{\phi }(\eta _ i^ l), \]
and $(\eta _ i^ k)_{k\geq 0}$ is a decreasing Cauchy sequence. Therefore, the monotone case in the proof of theorem 17.10.60 shows that
\[ \eta _ i\coloneqq \lim _{k\to \infty }\eta _ i^ k=\inf _{k\geq 0} \eta _ i^ k \]
is the $d_1$-limit of $(\eta _ i^ k)_{k\geq 0}$ and proposition 16.5.37 shows that
\[ d_1(\varphi _ i,\eta _ i)\leq 2^{1-i}. \]
By construction, $\eta _ i\leq \varphi _ i$ and $(\eta _ i)_ i$ is increasing. Therefore, $\varphi $ is the $d_1$-limit of the increasing sequence $(\eta _ i)_{i>0}$. As we have seen in the proof of theorem 17.10.60, this implies that $\varphi $ is also the almost everywhere limit of $(\eta _ i)_{i>0}$.
$\square$
Theorem 17.10.63.D84A6072F1A6 The functional $E_{\theta }^{\phi }\colon \mathcal{E}^1(X,\theta ;\phi ) \rightarrow \mathbb {R}$ is continuous.
Proof.
Let $(\varphi _ j)_{j>0}$ be a sequence in $\mathcal{E}^1(X,\theta ;\phi )$ with $d_1$-limit $\varphi \in \mathcal{E}^1(X,\theta ;\phi )$. We wish to show that
1C0180D1496E
17.10.64
\begin{equation} \lim _{j\to \infty } E_{\theta }^{\phi }(\varphi _ j)=E_{\theta }^{\phi }(\varphi ). \end{equation}
It suffices to prove that every subsequence has a further subsequence along which (17.10.64) holds. Thus, after replacing $(\varphi _ j)_{j>0}$ by a subsequence, let $(\eta _ j)_ j$ and $(\psi _ j)_ j$ be the monotone sequences supplied by 17.10.62. From the construction, $\eta _ j\leq \varphi _ j\leq \psi _ j$. By proposition 16.5.37,
\[ E_{\theta }^{\phi }(\eta _ j)\to E_{\theta }^{\phi }(\varphi ),\quad E_{\theta }^{\phi }(\psi _ j)\to E_{\theta }^{\phi }(\varphi ). \]
The monotonicity of $E_{\theta }^{\phi }$ then squeezes $E_{\theta }^{\phi }(\varphi _ j)$, proving (17.10.64).
$\square$
Proposition 17.10.65.0C8741A27906 Let $(\varphi _ t)_{t\in [a,b]}$ be a geodesic in $\mathcal{E}^1(X,\theta ;\phi )$. Then $t\mapsto E_{\theta }^{\phi }(\varphi _ t)$ is a linear function on $[a,b]$.
We recall the terminology used in the proof. Let $\Omega \subset \mathbb {C}^ m$ be Euclidean open. A function $u\in \mathrm{PSH}(\Omega )$ is
maximal if, for every open $G\Subset \Omega $ and every upper semicontinuous function $v$ on $\overline G$ such that $v\in \mathrm{PSH}(G)$ and $v\leq u$ on $\partial G$, one has $v\leq u$ on $G$. Let $\mathcal F$ denote the plurifine topology. If $U$ is $\mathcal F$-open and $u$ is $\mathcal F$-psh on $U$, then $u$ is
$\mathcal F$-maximal if, for every bounded $\mathcal F$-open set $G$ with $G\subset \overline G^{\mathcal F}\subset U$ and every bounded-above $\mathcal F$-upper semicontinuous function $v$ on $\overline G^{\mathcal F}$ that is $\mathcal F$-psh on $G$, one has
\[ v\leq u\textup{ on }\partial _{\mathcal F}G \quad \Longrightarrow \quad v\leq u\textup{ on }G. \]
Proof.
By an affine change of parameter, we may assume that $[a,b]=[0,1]$.
Step 1. Assume first that $\varphi _0,\varphi _1\in \mathcal{E}^{\infty }(X,\theta ;\phi )$. Let $Y=X\times \mathbb {P}^1$, and let $p_1,p_2$ be the two projections. On
\[ A\coloneqq \{ z\in \mathbb {C}:\mathrm{e}^{-1}<|z|<1\} , \qquad U(x,z)\coloneqq \varphi _{-\log |z|}(x), \]
the function $U$ is $p_1^*\theta $-psh. Choose $C>0$ such that
\[ \phi -C\leq \varphi _0,\varphi _1\leq \phi +C. \]
The constant subgeodesic at $\phi -C$ and proposition 17.10.7 give
ECB123C0AED3
17.10.66
\begin{equation} \phi -C\leq \varphi _ t\leq (1-t)\varphi _0+t\varphi _1\leq \phi +C, \qquad 0<t<1. \end{equation}
Fix concentric annuli $A_0\Subset A_1\Subset A$ and a unit-mass Fubini–Study form $\omega _{\mathrm{FS}}$ on $\mathbb {P}^1$. Put
\[ \beta =p_2^*\omega _{\mathrm{FS}},\qquad \Theta _ B=p_1^*\theta +B\beta ,\qquad \Phi =p_1^*\phi . \]
The class represented by $\Theta _ B$ is big. Since $\phi =P_\theta [\phi ]$, we have $\phi \leq 0$. For every $B>0$, $\Phi =P_{\Theta _ B}[\Phi ]$. Indeed, if $\Psi \in \mathrm{PSH}(Y,\Theta _ B)$ satisfies $\Psi \leq 0$ and $\Psi \preceq \Phi $, then on each fibre $X\times \{ z\} $, either $\Psi _ z\equiv -\infty $, or $\Psi _ z\in \mathrm{PSH}(X,\theta )$ is a candidate in the definition of $P_\theta [\phi ]=\phi $. Thus $\Psi _ z\leq \phi $ in both cases. Conversely, $\Phi $ itself is a candidate. Moreover, proposition 13.11.4 6 gives
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\begin{equation} \int _ Y(\Theta _ B+\mathrm{dd}^{\mathrm{c}}\Phi )^{n+1} =(n+1)B\int _ X\theta _\phi ^ n>0. \end{equation}
Choose an $S^1$-invariant smooth function $q$ on $\mathbb {P}^1$ such that
\[ q<-C-1\ \textup{ on }A_0, \qquad q>C+1\ \textup{ near }\partial A_1. \]
After increasing $B$, we may assume that $B\omega _{\mathrm{FS}}+\mathrm{dd}^{\mathrm{c}}q\geq 0$. Identifying $q$ with $p_2^*q$, define
\[ W= \begin{cases} U\lor (\Phi +q),& \textup{on }X\times A_1,\\ \Phi +q,& \textup{on }Y\setminus (X\times A_1). \end{cases} \]
The gluing lemma corollary 5.14.20 shows that $W\in \mathrm{PSH}(Y,\Theta _ B)$. By construction,
09167137EDA8
17.10.68
\begin{equation} W=U\ \textup{ on }X\times A_0, \qquad \Phi -C'\leq W\leq \Phi +C' \end{equation}
for some $C'>0$. Hence $W\in \mathcal{E}^{\infty }(Y,\Theta _ B;\Phi )$ by (17.10.67).
Set
\[ T=\Theta _ B+\mathrm{dd}^{\mathrm{c}}W,\qquad S=\Theta _ B+\mathrm{dd}^{\mathrm{c}}\Phi . \]
Define the order-zero current on $\mathbb {P}^1$
EC42C56857FF
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\begin{equation} \mathcal{H}_ W\coloneqq \frac{1}{n+1}(p_2)_*\left((W-\Phi )\sum _{j=0}^ n T^ j\wedge S^{n-j}\right). \end{equation}
The integration-by-parts formula [Xia19b], applied against a smooth test function on $\mathbb {P}^1$, gives
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\begin{equation} \mathrm{dd}^{\mathrm{c}}\mathcal{H}_ W =\frac{1}{n+1}(p_2)_*\left(T^{n+1}-S^{n+1}\right). \end{equation}
By (16.5.11), (17.10.68), and corollary 13.11.15, on $A_0$ the current $\mathcal{H}_ W$ is represented almost everywhere by
\[ z\longmapsto E_\theta ^\phi (\varphi _{-\log |z|}). \]
This representative is continuous. Indeed, on every smaller annulus, the convex functions $t\mapsto \varphi _ t(x)-\phi (x)$, considered on the common locus where both values are finite, are uniformly bounded by (17.10.66), hence uniformly Lipschitz. The resulting inequalities between the potentials hold almost everywhere and therefore everywhere by proposition 5.14.6. The monotonicity in 16.5.14 and (16.5.13) then give the local Lipschitz continuity of $t\mapsto E_\theta ^\phi (\varphi _ t)$.
On $X\times A_0$, define the signed Radon measure
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\begin{equation} \mu _ U\coloneqq \left(T^{n+1}-(n+1)B\beta \wedge T^ n\right)|_{X\times A_0}. \end{equation}
This measure is positive. To see this, take a relatively compact product coordinate ball $Q\subset X\times A_0$. Let $a$ be a smooth local potential of $p_1^*\theta $, put $h=U+a\leq 0$ and $h_ k=h\lor (-k)$, and choose a smooth local potential $b$ of $B\beta $ with $|b|\leq C_ Q$. For $p=n,n+1$, set
\[ \begin{aligned} \nu _ k^{b,p}& =\mathds {1}_{\{ h+b>-k\} } \bigl (\mathrm{dd}^{\mathrm{c}}((h+b)\lor (-k))\bigr )^ p,\\ \nu _ k^{h,p}& =\mathds {1}_{\{ h>-k\} } \bigl (\mathrm{dd}^{\mathrm{c}}h_ k+B\beta \bigr )^ p. \end{aligned} \]
Plurifine locality and $|b|\leq C_ Q$ give, for $k>C_ Q$,
\[ \nu _{k-C_ Q}^{h,p}\leq \nu _ k^{b,p}\leq \nu _{k+C_ Q}^{h,p}. \]
Thus the two cofinal truncation families have the same limit. By proposition 13.11.4 6 and $\beta ^2=0$,
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\begin{equation} \begin{aligned} & \mathds {1}_{\{ h>-k\} }\left[ (\mathrm{dd}^{\mathrm{c}}h_ k+B\beta )^{n+1} -(n+1)B\beta \wedge (\mathrm{dd}^{\mathrm{c}}h_ k+B\beta )^ n\right]\\ & \qquad =\mathds {1}_{\{ h>-k\} }(\mathrm{dd}^{\mathrm{c}}h_ k)^{n+1}\geq 0. \end{aligned} \end{equation}
Passing to the increasing truncation limit yields
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\begin{equation} \mu _ U|_ Q=\lim _{k\to \infty } \mathds {1}_{\{ h>-k\} }(\mathrm{dd}^{\mathrm{c}}h_ k)^{n+1}\geq 0. \end{equation}
By (13.11.17),
\[ (p_2)_*(\beta \wedge T^ n) =\left(\int _ X\theta _\phi ^ n\right)\omega _{\mathrm{FS}}, \]
since $W_ z\sim \phi $ on every fibre and proposition 16.3.10 applies. On the other hand,
\[ (p_2)_*S^{n+1} =(n+1)B\left(\int _ X\theta _\phi ^ n\right)\omega _{\mathrm{FS}} \]
by proposition 13.11.4 6. Together with (17.10.70) and (17.10.71), these identities give on $A_0$
0F481A9AC521
17.10.74
\begin{equation} \mathrm{dd}^{\mathrm{c}}_ z E_\theta ^\phi (\varphi _{-\log |z|}) =\frac{1}{n+1}(p_2)_*\mu _ U. \end{equation}
Step 2. We prove that $\mu _ U=0$. Let $D\Subset X\times A$ be a coordinate domain, let $a$ be a smooth local potential of $p_1^*\theta $ near $\overline D$, and set $h=U+a$. If $v$ is a psh function on $D$ with the boundary inequality required in the definition of maximality, max-pasting gives
\[ \widetilde U=U\lor (v-a)\ \textup{ on }D, \qquad \widetilde U=U\ \textup{ outside }D. \]
The function $\widetilde U$ need not be radial. Its $S^1$-symmetrization
\[ V(x,z)=\operatorname *{sup}_{|\lambda |=1}{}\! {}^{*}\widetilde U(x,\lambda z) \]
is radial and $p_1^*\theta $-psh. The $S^1$-saturation of $D$ is relatively compact in $X\times A$, so $V=U$ near $X\times \partial A$. Hence the family $V_ t(x)=V(x,\mathrm{e}^{-t})$ is a subgeodesic from $\varphi _0$ to $\varphi _1$ by definition 17.10.2. The definition (17.10.21) therefore gives
\[ \widetilde U\leq V\leq U. \]
It follows that $v\leq h$ on $D$, so $h$ is a maximal psh function.
On a product coordinate ball $Q\Subset X\times A_0$, put $O_ k=Q\cap \{ h>-k\} $. By [Proposition 5.9.3, EKF], $h$ is $\mathcal F$-maximal. The function $h_ k=h\lor (-k)$ is locally bounded and agrees with $h$ on the $\mathcal F$-open set $O_ k$, hence is $\mathcal F$-maximal there. Therefore [Theorem 5.10.1, EKF] gives
\[ \mathds {1}_{O_ k}(\mathrm{dd}^{\mathrm{c}}h_ k)^{n+1}=0. \]
By (17.10.73), $\mu _ U=0$. Since $A_0\Subset A$ was arbitrary, (17.10.74) shows that $z\mapsto E_\theta ^\phi (\varphi _{-\log |z|})$ is radial and harmonic on $A$. Thus $t\mapsto E_\theta ^\phi (\varphi _ t)$ is affine on $(0,1)$.
Choose $C_0>0$ such that $|\varphi _1-\varphi _0|\leq C_0$ on their common finite locus; equivalently, the two resulting inequalities between the potentials hold everywhere. Affine barriers and proposition 17.10.7 give
\[ \begin{aligned} \varphi _0-C_0t& \leq \varphi _ t\leq \varphi _0+C_0t,\\ \varphi _1-C_0(1-t)& \leq \varphi _ t\leq \varphi _1+C_0(1-t). \end{aligned} \]
By 16.5.14 and (16.5.13), the energy converges to the corresponding endpoint energy as $t\to 0+$ and $t\to 1-$. Hence
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17.10.75
\begin{equation} E_\theta ^\phi (\varphi _ t) =(1-t)E_\theta ^\phi (\varphi _0)+tE_\theta ^\phi (\varphi _1) \end{equation}
when the endpoints belong to $\mathcal E^\infty (X,\theta ;\phi )$.
Step 3. Let now $\varphi _0,\varphi _1\in \mathcal E^1(X,\theta ;\phi )$. By example 17.10.23 and (16.5.13), we may add different constants to the endpoints and assume that $\varphi _0,\varphi _1\leq \phi $. Put
\[ \varphi _ j^{(k)}=\varphi _ j\lor (\phi -k),\qquad j=0,1, \]
and let $(\varphi _ t^{(k)})_{t\in [0,1]}$ be the corresponding geodesic. The endpoint truncations decrease to $\varphi _ j$, and Perron monotonicity gives
\[ \varphi _ t\leq \varphi _ t^{(k+1)}\leq \varphi _ t^{(k)}. \]
Let $h_ t=\lim _{k\to \infty }\varphi _ t^{(k)}$. The decreasing limit of the complexifications is $p_1^*\theta $-psh and is not identically $-\infty $. Taking the limit in the convex upper bound gives, first almost everywhere and then everywhere by proposition 5.14.6,
64A3E8566CC1
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\begin{equation} \varphi _ t\leq h_ t\leq (1-t)\varphi _0+t\varphi _1. \end{equation}
Note that
\[ |h_ t-\varphi _0| \leq |\varphi _ t-\varphi _0|+t|\varphi _1-\varphi _0|\quad \textup{a.e.}. \]
Since $\varphi _ t\to \varphi _0$ in $L^1$ as $t\to 0+$ and $\varphi _0,\varphi _1\in L^1(X)$, it follows that $h_ t\to \varphi _0$ in $L^1$. The same argument at $t=1$ gives $h_ t\to \varphi _1$ in $L^1$. Together with the $p_1^*\theta $-psh complexification constructed above, these endpoint limits show by definition 17.10.2 that $(h_ t)$ is a subgeodesic from $\varphi _0$ to $\varphi _1$. Hence (17.10.21) gives $h_ t\leq \varphi _ t$. Together with (17.10.76), this proves
\[ \varphi _ t^{(k)}\searrow \varphi _ t,\qquad 0<t<1. \]
Applying proposition 16.5.37 at the endpoints and at each interior slice, and then using (17.10.75), gives
\[ \begin{aligned} E_\theta ^\phi (\varphi _ t) & =\lim _{k\to \infty }E_\theta ^\phi (\varphi _ t^{(k)})\\ & =\lim _{k\to \infty }\left((1-t)E_\theta ^\phi (\varphi _0^{(k)}) +tE_\theta ^\phi (\varphi _1^{(k)})\right)\\ & =(1-t)E_\theta ^\phi (\varphi _0)+tE_\theta ^\phi (\varphi _1). \end{aligned} \]
This concludes the proof.
$\square$
Proposition 17.10.77.7ADCF3A953E5 Let $(\varphi ^ i_0)_{i\in I}$, $(\varphi ^ i_1)_{i\in I}$ be two uniformly bounded from above increasing nets in $\mathcal{E}^{1}(X,\theta ;\phi )$. Let $(\varphi ^ i_ t)_{t\in (0,1)}$ be the geodesic from $\varphi _0^ i$ to $\varphi _1^ i$ for each $i\in I$. Then
\[ \left(\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi ^ i_ t\right)_{t\in (0,1)} \]
is the geodesic from $\operatorname *{sup}_{i}{}\! {}^{*} \varphi ^ i_0$ to $\operatorname *{sup}_{i}{}\! {}^{*} \varphi ^ i_1$.
Proof.
Put
\[ \varphi _ j=\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ j^ i\quad (j=0,1), \qquad h_ t=\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ t^ i, \]
and let $(g_ t)_{t\in (0,1)}$ be the geodesic from $\varphi _0$ to $\varphi _1$. By proposition 16.5.53, $\varphi _0,\varphi _1\in \mathcal E^1(X,\theta ;\phi )$. The slice families are uniformly bounded from above by proposition 17.10.7; hence $h_ t\in \mathcal E^1(X,\theta ;\phi )$ by proposition 16.5.53. Moreover, proposition 17.10.13 shows that $(h_ t)$ is a subgeodesic from $\varphi _0$ to $\varphi _1$. Thus
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\begin{equation} h_ t\leq g_ t. \end{equation}
For each $t$, the net $(\varphi _ t^ i)_{i\in I}$ is increasing. By proposition 5.14.5, its pointwise supremum agrees with $h_ t$ almost everywhere, and the analogous assertion holds at the endpoints. Therefore proposition 16.5.37, proposition 17.10.65 gives
\[ \begin{aligned} E_\theta ^\phi (h_ t) & =\lim _{i\in I}E_\theta ^\phi (\varphi _ t^ i)\\ & =\lim _{i\in I}\left((1-t)E_\theta ^\phi (\varphi _0^ i) +tE_\theta ^\phi (\varphi _1^ i)\right)\\ & =(1-t)E_\theta ^\phi (\varphi _0)+tE_\theta ^\phi (\varphi _1) =E_\theta ^\phi (g_ t). \end{aligned} \]
By (17.10.42), (17.10.78), and 17.10.45,
\[ d_1(h_ t,g_ t)=E_\theta ^\phi (g_ t)-E_\theta ^\phi (h_ t)=0, \]
so $h_ t=g_ t$ for all $t\in (0,1)$.
$\square$
Proposition 17.10.79.5408F4A1486E Let $(\varphi _ t)_{t\in [0,1]}$, $(\psi _ t)_{t\in [0,1]}$ be geodesics in $\mathcal{E}^1(X,\theta ;\phi )$. Then the distance $d_1(\varphi _ t,\psi _ t)$ is a convex function of $t\in [0,1]$.
Proof.
By definition of $d_1$ and proposition 17.10.65, it suffices to show the concavity of
\[ [0,1]\ni t\mapsto E_{\theta }^{\phi }(\varphi _ t\land \psi _ t). \]
Let $(\eta _ t)_{t\in [0,1]}$ be the geodesic from $\varphi _0\land \psi _0$ to $\varphi _1\land \psi _1$. Then for each $t\in [0,1]$, we have $\eta _ t\leq \varphi _ t\land \psi _ t$. Hence, by the monotonicity of $E_\theta ^\phi $ and proposition 17.10.65,
\[ E_{\theta }^{\phi }( \varphi _ t\land \psi _ t)\geq E_{\theta }^{\phi }(\eta _ t)=(1-t)E_{\theta }^{\phi }(\varphi _0\land \psi _0)+tE_{\theta }^{\phi }(\varphi _1\land \psi _1) \]
for each $t\in [0,1]$. Applying the same argument to the restrictions of $(\varphi _ t)_ t$ and $(\psi _ t)_ t$ to any subinterval of $[0,1]$, which are again geodesics by proposition 17.10.24, gives a similar inequality on every subinterval. Hence $t\mapsto E_\theta ^\phi (\varphi _ t\land \psi _ t)$ is concave, and our assertion follows.
$\square$
In particular, we can introduce:
Definition 17.10.80.6945CDF5DE8A Let $\ell ,\ell '\in \mathcal{R}^1(X,\theta ;\phi )$. We define
\[ d_1(\ell ,\ell ')\coloneqq \lim _{t\to \infty }\frac{1}{t}d_1(\ell _ t,\ell '_ t). \]
The limit exists in $[0,\infty ]$ by proposition 17.10.79.
Theorem 17.10.81.B30037E401C6 The function $d_1$ defined in definition 17.10.80 is a metric on $\mathcal{R}^1(X,\theta ;\phi )$.
Proof.
Symmetry and the triangle inequality follow from the corresponding pointwise properties of $d_1$ on $\mathcal{E}^1(X,\theta ;\phi )$.
We next show that $d_1(\ell ,\ell ')<\infty $ for any $\ell ,\ell '\in \mathcal{R}^1(X,\theta ;\phi )$. Let $C_\ell $ be the constant in proposition 17.10.39, and set $\tilde{\ell }_ t=\ell _ t-C_\ell t$. Then $\tilde{\ell }$ is again a geodesic ray, $\sup _ X\tilde{\ell }_ t=0$, and hence $\tilde{\ell }_ t\leq \phi $ by proposition 16.3.21. Thus, by proposition 17.10.65,
\[ d_1(\tilde{\ell }_ t,\phi )=-E_\theta ^\phi (\tilde{\ell }_ t)=-tE_\theta ^\phi (\tilde{\ell }_1). \]
Moreover, by (16.5.13), $d_1(\ell _ t,\tilde{\ell }_ t)=|C_\ell |t\int _ X\theta _{\phi }^ n$. Hence $d_1(\ell _ t,\phi )$ grows at most linearly in $t$. The same argument applies to $\ell '$, and the triangle inequality gives $d_1(\ell ,\ell ')<\infty $.
It remains to prove separation: suppose that $\ell ,\ell '\in \mathcal{R}^1(X,\theta ;\phi )$ and $d_1(\ell ,\ell ')=0$.
Fix $s>0$. Then it follows from proposition 17.10.79 that
\[ \frac{d_1(\ell _ s,\ell '_ s)}{s}\leq \lim _{t\to \infty }\frac{d_1(\ell _ t,\ell '_ t)}{t}=d_1(\ell ,\ell ')=0. \]
Therefore, $\ell _ s=\ell '_ s$ and hence $\ell =\ell '$.
$\square$
Definition 17.10.82.89AFFBB3978E We define the radial Monge–Ampère energy $\mathbf{E}^{\phi }\colon \mathcal{R}(X,\theta ;\phi )\rightarrow \mathbb {R}\cup \{ \pm \infty \} $ as follows:
\[ \mathbf{E}^{\phi }(\ell )\coloneqq \varlimsup _{t\to \infty }\frac{E_{\theta }^{\phi }(\ell _ t)}{t}. \]
When $\phi =V_{\theta }$, we write $\mathbf{E}$ instead of $\mathbf{E}^{V_{\theta }}$.
Proposition 17.10.83.3A33DCAAE061 Let $\ell ,\ell '\in \mathcal{R}^1(X,\theta ;\phi )$ and $\ell \leq \ell '$. Then
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\begin{equation} d_1(\ell ,\ell ')=\mathbf{E}^{\phi }(\ell ')-\mathbf{E}^{\phi }(\ell ). \end{equation}
Proof.
By proposition 17.10.65, the energy is linear along finite energy rays. Hence
\[ E_{\theta }^{\phi }(\ell _ t)=tE_{\theta }^{\phi }(\ell _1),\quad E_{\theta }^{\phi }(\ell '_ t)=tE_{\theta }^{\phi }(\ell '_1). \]
Since $\ell _ t\leq \ell '_ t$ for all $t\geq 0$, (17.10.42) gives
\[ \frac{1}{t}d_1(\ell _ t,\ell '_ t)=\frac{1}{t}\left(E_{\theta }^{\phi }(\ell '_ t)-E_{\theta }^{\phi }(\ell _ t)\right). \]
Letting $t\to \infty $ proves the assertion.
$\square$
Next we recall the $\lor $ operator at the level of geodesic rays.
Definition 17.10.85.F06EA26AA808 Let $\ell ,\ell '\in \mathcal{R}^{\infty }(X,\theta ;\phi )$. We define $\ell \lor \ell '$ as the minimal ray in $\mathcal{R}^{\infty }(X,\theta ;\phi )$ lying above both $\ell $ and $\ell '$.
Proposition 17.10.86.B40B93A4363D Given $\ell ,\ell '\in \mathcal{R}^{\infty }(X,\theta ;\phi )$. Then $\ell \lor \ell '\in \mathcal{R}^{\infty }(X,\theta ;\phi )$ exists, and
CA6EEFF029A5
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\begin{equation} \mathbf{E}^{\phi }(\ell \lor \ell ')=\lim _{t\to \infty }\frac{1}{t}E_{\theta }^{\phi }\left(\ell _ t\lor \ell '_ t\right). \end{equation}
Proof.
For each $t>0$, let $(\ell ''^ t_{s})_{s\in [0,t]}$ be the geodesic from $\phi $ to $\ell _ t\lor \ell '_ t$.
Step 1. We first show that for each fixed $s\geq 0$, $\ell ''^ t_{s}$ is increasing in $t\in [s,\infty )$.
To see this, fix $s\geq 0$ and choose $t'>t\geq s$. We need to show that
947F65850DBC
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\begin{equation} \ell ''^{t'}_ s\geq \ell ''^{t}_ s. \end{equation}
Since $(\ell ''^{t'}_ a)_{a\in [0,t]}$ is a geodesic, it suffices to show that $(\ell ''^ t_ a)_{a\in [0,t]}$ is a candidate in the Perron envelope defining the former geodesic. In other words, in verifying (17.10.88), we may assume that either $s=0$ or $s=t$. The case $s=0$ is of course trivial. So it remains to prove the following:
\[ \ell ''^{t'}_ t\geq \ell _ t\lor \ell '_ t. \]
By symmetry, it suffices to prove
\[ \ell ''^{t'}_ t\geq \ell _ t. \]
But since $(\ell _ a)_{a\in [0,t']}$ is a candidate in the Perron envelope defining $\ell ''^{t'}$, this inequality follows.
Step 2. Next, observe that for a fixed $s\geq 0$, we have
\[ \sup _ X \ell ''^ t_ s\leq \frac{s}{t}\sup _ X\ell ''^ t_ t+ \frac{t-s}{t}\sup _ X \ell ''^ t_0=\frac{s}{t}\left( \sup _ X\ell _ t \right)\lor \left( \sup _ X\ell '_ t \right) \]
for all $t>s$. The right-hand side is bounded from above by a constant independent of $t\geq s$ by proposition 17.10.39. Let
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\begin{equation} (\ell \lor \ell ')_ s\coloneqq \operatorname *{sup}_{t>s}{}\! {}^{*} \ell ''^ t_{s}. \end{equation}
Then proposition 17.10.77 guarantees that $\ell \lor \ell '\in \mathcal{R}^{\infty }(X,\theta ;\phi )$.
Step 3. We need to show that $\ell \lor \ell '$ defined in this way is indeed the minimal ray lying above $\ell $ and $\ell '$.
First, by Step 1, we have
\[ \ell ''^ t_ s\geq \ell ''^ s_ s\geq \ell _ s \]
for any $t\geq s\geq 0$. Therefore,
\[ (\ell \lor \ell ')_ s\geq \ell _ s \]
for all $s\geq 0$. In other words, $\ell \lor \ell '\geq \ell $. Similarly, $\ell \lor \ell '\geq \ell '$.
Next, let $L\in \mathcal{R}^{\infty }(X,\theta ;\phi )$ be a ray lying above both $\ell $ and $\ell '$. Then we have
\[ L_ t\geq \ell _ t\lor \ell '_ t \]
for all $t\geq 0$. In particular,
\[ L_ s\geq \ell ''^ t_ s \]
for all $t\geq s\geq 0$. It follows that
\[ L_ s\geq (\ell \lor \ell ')_ s \]
for all $s\geq 0$.
Step 4. It remains to argue (17.10.87). Since the energy is linear along bounded rays,
\[ \mathbf{E}^{\phi }(\ell \lor \ell ')=E_{\theta }^{\phi }\left((\ell \lor \ell ')_1\right). \]
Moreover, $\ell ''^ t_1$ increases to $(\ell \lor \ell ')_1$ as $t\to \infty $. By proposition 16.5.37,
\[ E_{\theta }^{\phi }\left((\ell \lor \ell ')_1\right)=\lim _{t\to \infty } E_{\theta }^{\phi }\left(\ell ''^ t_1\right). \]
Finally, by proposition 17.10.65 applied to $(\ell ''^ t_ s)_{s\in [0,t]}$,
\[ E_{\theta }^{\phi }\left(\ell ''^ t_1\right)=\frac{1}{t}E_{\theta }^{\phi }\left(\ell _ t\lor \ell '_ t\right). \]
Combining these identities gives (17.10.87).
$\square$
Lemma 17.10.90.6C6200F67BD7 For any $\ell ,\ell '\in \mathcal{R}^{\infty }(X,\theta ;\phi )$, we have
FE9D22CC2BA6
17.10.91
\begin{equation} d_1(\ell ,\ell ')\leq d_1(\ell ,\ell \lor \ell ')+d_1(\ell ',\ell \lor \ell ')\leq C_ nd_1(\ell ,\ell '), \end{equation}
where $C_ n=3(n+1)2^{n+2}$.
Proof.
The first inequality is trivial. As for the second, we estimate
\[ \begin{aligned} d_1(\ell ,\ell \lor \ell ')=& \mathbf{E}^{\phi }(\ell \lor \ell ')-\mathbf{E}^{\phi }(\ell )& \textup{by proposition \href{../../tag/3A33DCAAE061/}{17.10.83}}\\ =& \lim _{t\to \infty } \frac{1}{t}E_{\theta }^{\phi }\left(\ell _ t\lor \ell '_ t\right)-\mathbf{E}^{\phi }(\ell )& \textup{by (\href{../../tag/CA6EEFF029A5/}{17.10.87})}\\ =& \lim _{t\to \infty } \frac{1}{t} d_1\left(\ell _ t\lor \ell '_ t,\ell _ t\right)& \textup{by (\href{../../tag/D8C8C0BB6585/}{17.10.42})}. \end{aligned} \]
By symmetry, we find
\[ d_1(\ell ,\ell \lor \ell ')+d_1(\ell ',\ell \lor \ell ')\leq \lim _{t\to \infty } \frac{1}{t} \Bigl (d_1\left(\ell _ t\lor \ell '_ t,\ell _ t\right)+d_1\left(\ell _ t\lor \ell '_ t,\ell '_ t\right)\Bigr ). \]
By theorem 17.10.48 and 17.10.47, for each $t>0$,
\[ d_1\left(\ell _ t\lor \ell '_ t,\ell _ t\right)+d_1\left(\ell _ t\lor \ell '_ t,\ell '_ t\right)\leq 3(n+1)2^{n+2} d_1(\ell _ t,\ell '_ t). \]
Now (17.10.91) follows.
$\square$
Example 17.10.92.02BD88048323 Let $\varphi \in \mathrm{PSH}(X,\theta )$. Assume that $\varphi \leq 0$. For each $C>0$, let $(\ell ^{\varphi ,C}_ t)_{t\in [0,C]}$ be the geodesic from $V_{\theta }$ to $(V_{\theta }-C)\lor \varphi $. For each $t\geq 0$, there is $\ell ^{\varphi }_ t\in \mathcal{E}^{\infty }(X,\theta )$ such that
BF8F51ADD4CB
17.10.93
\begin{equation} \ell ^{\varphi ,C}_ t\xrightarrow {d_1}\ell ^{\varphi }_ t \end{equation}
as $C\to \infty $. Then $\ell ^{\varphi }\in \mathcal{R}^{\infty }(X,\theta )$ and
795CA33E2D29
17.10.94
\begin{equation} \mathbf{E}(\ell ^{\varphi })=\frac{1}{n+1} \sum _{j=0}^ n \left(\int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{V_{\theta }}^ n\right). \end{equation}
Comparing the above approximants with cutoff parameters shifted by a bounded amount shows that $\ell ^{\varphi +C}=\ell ^{\varphi }$ for any $C\in \mathbb {R}$. Therefore, for a general $\varphi \in \mathrm{PSH}(X,\theta )$, we may simply define
\[ \ell ^{\varphi }\coloneqq \ell ^{\varphi -\sup _ X\varphi }. \]
Then the conclusions of this example continue to hold.
Proof.
We first show that for each fixed $t\geq 0$, $\ell _ t^{\varphi ,C}$ is increasing in $C\geq t$.
To see this, choose $t\leq C_1<C_2$. We need to show that
\[ \ell ^{\varphi ,C_1}_ t\leq \ell ^{\varphi ,C_2}_ t. \]
Since both sides are geodesics for $t\in [0,C_1]$, it suffices to show that
277396283395
17.10.95
\begin{equation} (V_{\theta }-C_1)\lor \varphi \leq \ell _{C_1}^{\varphi ,C_2}. \end{equation}
Now $((V_{\theta }-t)\lor \varphi )_{t\in [0,C_2]}$ is a subgeodesic from $V_{\theta }$ to $(V_{\theta }-C_2)\lor \varphi $ by proposition 17.10.13. It is therefore a candidate in the Perron envelope defining $\ell ^{\varphi ,C_2}$, and evaluating at $t=C_1$ gives exactly (17.10.95).
From proposition 17.10.7, we know that for any $C>t>0$, we have
\[ \ell ^{\varphi ,C}_ t\leq \frac{t}{C}\Bigl ((V_{\theta }-C)\lor \varphi \Bigr )+\frac{C-t}{C}\cdot V_{\theta }\leq 0, \]
while the linear subgeodesic $V_\theta -t$ is a candidate in the same Perron envelope, so $V_\theta -t\leq \ell ^{\varphi ,C}_ t$. Therefore, by proposition 5.14.2,
AAACF2ED8350
17.10.96
\begin{equation} \ell _ t^{\varphi }\coloneqq \operatorname *{sup}_{C>t}{}\! {}^{*}\ell ^{\varphi ,C}_ t\in \mathcal{E}^{\infty }(X,\theta ) \end{equation}
for all $t\geq 0$. Thanks to proposition 16.5.37, we have
\[ \ell ^{\varphi ,C}_ t\xrightarrow {d_1} \ell _ t^{\varphi } \]
as $C\to \infty $ for all $t\geq 0$. Applying proposition 17.10.77 on each interval $[0,T]$ to the increasing family $(\ell ^{\varphi ,C}_ t)_{t\in [0,T]}$, $C\geq T$, shows that $\ell ^{\varphi }\in \mathcal{R}^{\infty }(X,\theta )$.
It remains to compute the energy of $\ell ^{\varphi }$. We first fix $C\geq t>0$ and compute using proposition 17.10.65:
\[ E_{\theta }\left(\ell ^{\varphi ,C}_ t\right)=\frac{t}{C}E_{\theta }\Bigl ((V_{\theta }-C)\lor \varphi \Bigr ). \]
Letting $C\to \infty $ and applying proposition 16.5.37, we find that
\[ E_{\theta }(\ell ^{\varphi }_ t)=\lim _{C\to \infty }\frac{t}{C}E_{\theta }\Bigl ((V_{\theta }-C)\lor \varphi \Bigr ) \]
for any $t\geq 0$. It follows that
\[ \mathbf{E}(\ell ^{\varphi })=\lim _{C\to \infty }\frac{1}{C}E_{\theta }\Bigl ((V_{\theta }-C)\lor \varphi \Bigr ). \]
Using the definition of $E_{\theta }$, in order to obtain (17.10.94), it suffices to show that for each $j=0,\ldots ,n$, we have
287169AF175C
17.10.97
\begin{equation} \lim _{C\to \infty }\int _ X \frac{(V_{\theta }-C)\lor \varphi -V_{\theta }}{C} \, \theta _{(V_{\theta }-C)\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\varphi }^{j}\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{V_{\theta }}^ n. \end{equation}
For this purpose, for each $C>0$, we decompose $X$ as $\{ \varphi > V_{\theta }-C\} $ and $\{ \varphi \leq V_{\theta }-C\} $. We have
\[ \begin{aligned} & \int _{\{ \varphi > V_{\theta }-C\} } \frac{(V_{\theta }-C)\lor \varphi -V_{\theta }}{C} \, \theta _{(V_{\theta }-C)\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}\\ =& \int _{\{ \varphi > V_{\theta }-C\} } \frac{\varphi -V_{\theta }}{C} \, \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \end{aligned} \]
On the other hand,
\[ \begin{aligned} & \int _{\{ \varphi \leq V_{\theta }-C\} } \frac{(V_{\theta }-C)\lor \varphi -V_{\theta }}{C} \, \theta _{(V_{\theta }-C)\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}\\ =& -\int _{\{ \varphi \leq V_{\theta }-C\} } \theta _{(V_{\theta }-C)\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}\\ =& -\int _ X \theta _{V_{\theta }}^ n+\int _{\{ \varphi > V_{\theta }-C\} } \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \end{aligned} \]
Observe that for $C>0$, the function $\mathds {1}_{\{ \varphi >V_{\theta }-C\} }C^{-1}(\varphi -V_{\theta })$ is defined quasi-everywhere. Since $\varphi \leq V_\theta $, it takes values in $[-1,0]$ outside a pluripolar set. When $C\to \infty $, these functions converge to $0$ quasi-everywhere, so the first integral above converges to $0$ by dominated convergence. Similarly, the indicators $\mathds {1}_{\{ \varphi >V_\theta -C\} }$ increase to $1$ outside a pluripolar set, hence
\[ \lim _{C\to \infty }\int _{\{ \varphi >V_{\theta }-C\} } \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j} = \int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \]
Therefore, (17.10.97) follows.
$\square$
Pome
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